We consider universal computability of the STRS with no additional rule schemes:

  Signature: cons    :: c -> d -> d
             f       :: a -> a -> a
             false   :: b
             filter  :: (c -> b) -> d -> d
             filter2 :: b -> (c -> b) -> c -> d -> d
             g       :: a -> a
             h       :: a -> a
             map     :: (c -> c) -> d -> d
             nil     :: d
             true    :: b

  Rules: g(h(g(X))) -> g(X)
         g(g(Y)) -> g(h(g(Y)))
         h(h(U)) -> h(f(h(U), U))
         map(H, nil) -> nil
         map(I, cons(P, X1)) -> cons(I(P), map(I, X1))
         filter(Z1, nil) -> nil
         filter(G1, cons(V1, W1)) -> filter2(G1(V1), G1, V1, W1)
         filter2(true, J1, X2, Y2) -> cons(X2, filter(J1, Y2))
         filter2(false, G2, V2, W2) -> filter(G2, W2)

The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the initial DP problem D1 = (P1, R UNION R_?, i, c), where:

  P1. (1)  g#(h(g(X))) => g#(X)
      (2)  g#(g(Y)) => g#(Y)
      (3)  g#(g(Y)) => h#(g(Y))
      (4)  g#(g(Y)) => g#(h(g(Y)))
      (5)  h#(h(U)) => h#(U)
      (6)  h#(h(U)) => h#(f(h(U), U))
      (7)  map#(I, cons(P, X1)) => map#(I, X1)
      (8)  filter#(G1, cons(V1, W1)) => filter2#(G1(V1), G1, V1, W1)
      (9)  filter2#(true, J1, X2, Y2) => filter#(J1, Y2)
      (10) filter2#(false, G2, V2, W2) => filter#(G2, W2)

***** We apply the Graph Processor on D1 = (P1, R UNION R_?, i, c).

We compute a graph approximation with the following edges:

  1:  1 2 3 4 
  2:  1 2 3 4 
  3:  5 6 
  4:  1 2 3 4 
  5:  5 6 
  6:
  7:  7 
  8:  9 10 
  9:  8 
  10: 8 

There are 4 SCCs.

Processor output: { D2 = (P2, R UNION R_?, i, c) ; D3 = (P3, R UNION R_?, i, c) ; D4 = (P4, R UNION R_?, i, c) ; D5 = (P5, R UNION R_?, i, c) }, where:

  P2. (1) h#(h(U)) => h#(U)

  P3. (1) g#(h(g(X))) => g#(X)
      (2) g#(g(Y)) => g#(Y)
      (3) g#(g(Y)) => g#(h(g(Y)))

  P4. (1) map#(I, cons(P, X1)) => map#(I, X1)

  P5. (1) filter#(G1, cons(V1, W1)) => filter2#(G1(V1), G1, V1, W1)
      (2) filter2#(true, J1, X2, Y2) => filter#(J1, Y2)
      (3) filter2#(false, G2, V2, W2) => filter#(G2, W2)

***** We apply the Subterm Criterion Processor on D2 = (P2, R UNION R_?, i, c).

We use the following projection function:

  nu(h#) = 1

We thus have:

  (1) h(U) |>| U

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.

***** We apply the Usable Rules Processor on D3 = (P3, R UNION R_?, i, c).

We obtain 3 usable rules (out of 9 rules in the input problem).

Processor output: { D6 = (P3, R2, i, c) }, where:

  R2. (1) g(h(g(X))) -> g(X)
      (2) g(g(Y)) -> g(h(g(Y)))
      (3) h(h(U)) -> h(f(h(U), U))

***** We apply the Subterm Criterion Processor on D4 = (P4, R UNION R_?, i, c).

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(P, X1) |>| X1

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.

***** We apply the Subterm Criterion Processor on D5 = (P5, R UNION R_?, i, c).

We use the following projection function:

  nu(filter#)  = 2
  nu(filter2#) = 4

We thus have:

  (1) cons(V1, W1) |>|  W1
  (2) Y2           |>=| Y2
  (3) W2           |>=| W2

We may remove the strictly oriented DPs.

Processor output: { D7 = (P6, R UNION R_?, i, c) }, where:

  P6. (1) filter2#(true, J1, X2, Y2) => filter#(J1, Y2)
      (2) filter2#(false, G2, V2, W2) => filter#(G2, W2)

***** No progress could be made on DP problem D6 = (P3, R2, i, c).